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IMPLEMENTATION OF COUPLED THERMAL AND STRUCTURAL ANALYSIS
METHODS FOR REINFORCED CONCRETE STRUCTURES
A THESIS SUBMITTED TO
THE GRADUATE SCHOOL OF NATURAL AND APPLIED SCIENCES
OF
MIDDLE EAST TECHNICAL UNIVERSITY
BY
UTKU ALBOSTAN
IN PARTIAL FULFILLMENT OF THE REQUIREMENTS
FOR
THE DEGREE OF MASTER SCIENCE IN
CIVIL ENGINEERING
JANUARY 2013
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Approval of the thesis:
IMPLEMENTATION OF COUPLED THERMAL AND STRUCTURAL ANALYSIS
METHODS FOR REINFORCED CONCRETE STRUCTURES
submitted by UTKU ALBOSTAN in partial fulfillment of the requirements for the degree of Master of Science in Civil Engineering Department, Middle East
Technical University by,
Prof. Dr. Canan Özgen
Dean, Graduate School of Natural and Applied Sciences
Prof. Dr. Ahmet Cevdet Yalçıner
Head of Department, Civil Engineering
Assoc. Prof. Dr. Özgür Kurç
Supervisor, Civil Engineering Dept., METU
Examining Committee Members:
Prof. Dr. Ahmet Yakut
Civil Engineering Dept., METU
Assoc. Prof. Dr. Özgür Kurç
Civil Engineering Dept., METU
Prof. Dr. Barış Binici
Civil Engineering Dept., METU
Assoc. Prof. Dr. Yalın Arıcı
Civil Engineering Dept., METU
Assoc. Prof. Dr. Görkem Külah
Chemical Engineering Dept., METU
DATE:
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I hereby declare that all information in this document has been obtained and
presented in accordance with academic rules and ethical conduct. I also declare that, as required by these rules and conduct, I have fully cited and
referenced all material and results that are not original to this work.
Name, Last name : UTKU ALBOSTAN
Signature :
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ABSTRACT
IMPLEMENTATION OF COUPLED THERMAL AND STRUCTURAL ANALYSIS
METHODS FOR REINFORCED CONCRETE STRUCTURES
Albostan, Utku M.S, Department of Civil Engineering
Supervisor : Assoc. Prof. Dr. Özgür Kurç
January 2013, 105 pages
Temperature gradient causes volume change (elongation/shortening) in concrete
structures. If the movement of the structure is restrained, significant stresses may
occur on the structure. These stresses may be so significant that they can cause
considerable cracking at structural components of large concrete structures. Thus, during the design of a concrete structure, the actual temperature gradient in the
structure should be obtained in order to compute the stress distribution on the
structure due to thermal effects. This study focuses on the implementation of a
solution procedure for coupled thermal and structural analysis with finite element
method for such structures. For this purpose, first transient heat transfer analysis algorithm is implemented to compute the thermal gradient occurring inside the
concrete structures. Then, the output of the thermal analysis is combined with the
linear static solution algorithm to compute stresses due to temperature gradient.
Several, 2D and 3D, finite elements having both structural and thermal analysis
capabilities are developed. The performances of each finite element are investigated.
As a case study, the top floor of two L-shaped reinforced concrete parking structure and office building are analyzed. Both structures are subjected to heat convection
at top face of the slabs as ambient condition. The bottom face of the slab of the
parking structure has the same thermal conditions as the top face whereas in the
office building the temperature inside the building is fixed to 20 degrees. The
differences in the stress distribution of the slabs and the internal forces of the
vertical structural members are discussed.
Keywords: Finite Element, Heat Transfer Analysis, Coupled Analysis, Thermal
Gradient, Reinforced Concrete Structure.
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ÖZ
BETONARME YAPILAR İÇİN ISI İLETİMİ VE YAPISAL ÇÖZÜMLEME METOTLARI
KULLANILARAK İKİLİ ÇÖZÜMLEME YÖNTEMİNİN GELİŞTİRİLMESİ
Albostan, Utku Yüksek Lisans, İnşaat Mühendisliği Bölümü
Tez Yöneticisi : Doç. Dr. Özgür Kurç
Ocak 2013, 105 sayfa
Betonarme yapılarda sıcaklık değişimi hacim değişikliğine (genleşme/büzülme)
sebep olmaktadır. Eğer yapının eksenel yöndeki deformasyonu engellenirse yapı
üzerinde ciddi mertebede gerilmelerin oluştuğu görülmektedir. Bu yüksek
gerilmeler büyük betonarme yapılarda kırılma veya çatlamalara sebep olabilmektedir. Bu nedenle betonarme yapıların tasarımı sırasında, yapısal
elemanlarda sıcaklık farklılıkları nedeniyle oluşan gerilme miktarlarının
hesaplanabilmesi için, elemanlardaki sıcaklık dağılımı dikkate alınmalıdır. Bu
çalışma betonarme yapıların sonlu elemanlar yöntemi ile geliştirilmiş ısı iletimi ve
yapısal çözümleme yöntemleri ile ikili olarak çözümlenmesini incelemektedir. Bu amaçla ilk olarak zamana bağlı ısı iletimi çözümlemesi yapılarak betonarme
elemanların içerisinde oluşan sıcaklık dağılımı elde edilmiştir. Sonrasında bu
sıcaklık değerleri kullanılarak sistem yapısal olarak çözümlenmiş ve yapıdaki
gerilme değerleri hesaplanmıştır. Bu çözümleme uygulamasında kullanılmak üzere
2 ve 3 boyutlu çeşitli sonlu elemanlar geliştirilmiştir. Bu elemanlar hem ısı iletimi
hem de yapısal çözümlemelerde kullanılabilecek şekilde geliştirilmiş ve doğrulama testleri yapılmıştır. Test problemi olarak L şeklindeki betonarme park yeri yapısı ile
ofis binasının en üst katları çözümlenmiştir. Her iki yapının çatı döşemesinin üst
yüzeyine ısı konveksiyonu uygulanmıştır. Park yeri binasının çatı döşemesinin alt
yüzeyine de üst yüzeye uygulanan ısı yükü aynen etki ettirilirken ofis binasının iç
sıcaklığı 20 derecede sabit tutulmuştur. Her iki yapıda sıcaklık yükleri nedeniyle çatı döşemelerinde oluşan gerilmeler ve düşey elemanlarda oluşan iç kuvvetler
karşılaştırılmıştır.
Anahtar Kelimeler: Sonlu Eleman, Isı İletimi Çözümlemesi, İkili Çözümleme,
Sıcaklık Değişimi, Betonarme Yapılar.
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To My Parents
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ACKNOWLEDGEMENTS
I wish to express my utmost gratitude to my supervisor Assoc. Prof. Dr. Özgür Kurç for his guidance, advice, criticism, support, encouragements and insight
throughout the research.
Assist. Prof. Dr. Serdar Göktepe is also sincerely acknowledged for his valuable
support.
I am grateful for all continuous support, understanding and encouragement I have received from all of my family members, my mother Seval Albostan, my father, Erol
Albostan, and my sister Duygu Albostan Yenerim and her husband Cengizhan
Yenerim.
I would like to express sincere appreciation to my project members Tunç
Bahçecioğlu and Tolga Kurt. I would also like to appreciate my friends for standing
next to me during this process: Ali Baykara, Ezgi Berberoğlu, Başar Ataman, Aysim Damla Atalay, Muhammed Bulut, Oğuz Barın, Fırat Özel, Yunus Avcı, Hayati
Arslan, Sadettin Uğur, Mehmet Tuna, and Mehmet Kemal Ardoğa. Finally, I would
like to express my gratitude to Feyza Soysal for her continuous support and
endless patience throughout this study and in all aspects of my life.
This study has been conducted with the funding provided by The Scientific and Technological Research Council of Turkey (TUBITAK) under the grant
MAG108M586.
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TABLE OF CONTENTS
ABSTRACT ........................................................................................................... iv ÖZ ........................................................................................................................ v ACKNOWLEDGEMENTS ...................................................................................... vii TABLE OF CONTENTS ....................................................................................... viii LIST OF FIGURES ................................................................................................ x LIST OF TABLES ................................................................................................. xii
CHAPTERS
1.INTRODUCTION ................................................................................................ 1 1.1. Problem Definition .................................................................................. 1 1.2. Related Work .......................................................................................... 2 1.3. Objectives and Scopes ............................................................................ 6 1.4. Thesis Outline ........................................................................................ 6
2.THEORY ........................................................................................................... 9 2.1. Introduction ........................................................................................... 9 2.2. Structural Analysis ................................................................................. 9 2.3. Heat Transfer Analysis .......................................................................... 12 2.4. Coupled Analysis Methods .................................................................... 15
3.IMPLEMENTATION .......................................................................................... 17 3.1. Introduction ......................................................................................... 17 3.2. Structure of Panthalassa ...................................................................... 17 3.3. Solution Algorithms .............................................................................. 18
3.3.1. Linear Static Analysis .................................................................... 18 3.3.2. Linear Heat Transfer Analysis ........................................................ 19 3.3.3. Coupled Analysis ........................................................................... 23
3.3.4. Parallel Solution Algorithms………………………………………………… 26
3.4. Finite Elements .................................................................................... 28 3.4.1. Geometrical Properties of Finite Elements....................................... 28 3.4.2. Numerical Integration .................................................................... 33
4.VERIFICATION PROBLEMS ............................................................................. 37 4.1. Introduction ......................................................................................... 37 4.2. Structural Analysis Verification ............................................................ 37
4.2.1. Linear Static Problems ................................................................... 37 4.2.2. Temperature Load .......................................................................... 50
4.3. Verification of Heat Transfer Analysis .................................................... 53 4.3.1. Square Plate Problem ..................................................................... 53 4.3.2. Rectangular Plate Problem ............................................................. 56
4.4. SUMMARY............................................................................................ 60 5.CASE STUDY .................................................................................................. 61
5.1. Introduction ......................................................................................... 61 5.2. Model Properties ................................................................................... 61 5.3. Case 1: Parking Structure ..................................................................... 65 5.4. Case 2: Office Structure ........................................................................ 75 5.5. Comparison .......................................................................................... 84 5.6. Parallelization Aspect ............................................................................ 88
6.CONCLUSION AND FUTURE PLANS ................................................................ 91 6.1. Conclusion ........................................................................................... 91 6.2. Future Plans ........................................................................................ 92
REFERENCES .................................................................................................... 95
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APPENDICES A.INTEGRATION POINTS .................................................................................... 97
A.1. Integration Points for Line Element ....................................................... 97 A.2. Integration Points for a Quadrilateral Element ....................................... 97 A.3. Integration Points for Triangular Element .............................................. 98 A.4. Integration Points for Hexahedral Element ............................................ 99 A.5. Integration Points for Wedge Element .................................................... 99 A.6. Integration Points for Tetrahedral Element .......................................... 100
B.INPUT FORMAT OF PANTHALASSA ............................................................... 101 C.OUTPUT FORMAT OF PANTHALASSA............................................................ 105
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LIST OF FIGURES
FIGURES
Figure 2-1 Control Volume (Lewis et al.,2004) ..................................................... 13 Figure 3-1 Connections of Plug-ins with Panthalassa Engine ............................... 18
Figure 3-2 Flow Chart of Linear Static Analysis Algorithm .................................... 19
Figure 3-3 Flow Chart of Matrix Assembly of Heat Transfer Solution Algorithm ..... 20
Figure 3-4 Flow Chart of Linear Steady-State Analysis Algorithm ........................ 21 Figure 3-5 Flow Chart of Linear Transient Heat Transfer Analysis Algorithm ....... 23 Figure 3-6 Flow Chart of Coupled Analysis Algorithms with Transient Solution in
Weak Form ........................................................................................... 24 Figure 3-7 Flow Chart of Coupled Analysis Algorithm with Steady-State Solution in
Weak Form ........................................................................................... 25 Figure 3-8 Flow Chart of Converting Nodal Temperatures to Equivalent Nodal Force
............................................................................................................ 25 Figure 3-9 Flow Chart of Parallel Steady-State Heat Transfer Algorithm ................ 26
Figure 3-10 Flow Chart of Parallel Transient Heat Transfer Analysis Algorithm ..... 27
Figure 4-1 Straight Beams with Quadrilateral Elements. (A) Rectangular Meshed
Beam. (B) Trapezoidal Meshed Beam. .................................................... 38 Figure 4-2 Straight Beams with Triangular Elements. (D) With 12 Triangular
Elements. (E) With 24 Triangular Elements ........................................... 38 Figure 4-3 Straight Beams with Hexahedral Elements. (A) Rectangular Prismatic
Element. (B) Trapezoidal Prismatic Element. (C) Parallelogram Prismatic
Element ................................................................................................ 42 Figure 4-4 Straight Beams with Wedge Elements. (D) Beam with Linear Wedge
Elements. (E) Beam with Quadratic Wedge Elements ............................. 42 Figure 4-5 Cross Section of Beam with Linear Edge Elements ............................. 47 Figure 4-6 Plate Models with Quadrilateral Elements .......................................... 50 Figure 4-7 Plate Models with Triangular Elements .............................................. 51 Figure 4-8 3D Temperature Load ........................................................................ 52 Figure 4-9 Plate Models. (A) Full Plate. (B) Quarter Plate ..................................... 54 Figure 4-10 Temperature Distribution of Full and Quarter Plate .......................... 54 Figure 4-11 Rectangular Plate Model .................................................................. 56 Figure 4-12 Temperature Distribution of Rectangular Plate Modeled with TriM3
Elements .............................................................................................. 57 Figure 4-13 Rectangular Plate Modeled with 1000 TriM3 Elements ...................... 59 Figure 5-1 Plan View of L-Shaped Structure........................................................ 62 Figure 5-2 Section Properties of Structural Elements .......................................... 62
Figure 5-3 3D Model of L-Shaped Structure ......................................................... 63
Figure 5-4 Model Mesh ....................................................................................... 63 Figure 5-5 Hourly Temperature Distribution of Adana ......................................... 65
Figure 5-6 Section Cut of the Slab and Point 1 ..................................................... 66
Figure 5-7 Temperature Distribution of Slab of Parking Structure (A) 5th hour (B) 12th hour (C) 14th hour (D) 20th hour (E) 24th hour .................................. 67
Figure 5-8 Temperature Gradient through Thickness ........................................... 68
Figure 5-9 Mean Temperature Distribution of Slab Layers (Parking Structure) ...... 69
Figure 5-10 Displacements at 24th Hour. (A) x Direction (B) y Direction ................ 69
Figure 5-11 Stress Distributions of Slab at 5th Hour (Parking Structure) (MPa). (A) Stresses in x Direction. (B) Stresses in y Direction ................................. 70
Figure 5-12 Stress Distributions of Slab at 12th Hour (Parking Structure) (MPa). (A)
Stresses in x Direction. (B) Stresses in y Direction ................................. 71
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Figure 5-13 Stress Distributions of Slab at 14th Hour (Parking Structure) (MPa). (A)
Stresses in x Direction (B). Stresses in y Direction ................................. 72 Figure 5-14 Stress Distributions of Slab at 20th Hour (Parking Structure) (MPa). (A)
Stresses in x Direction (B). Stresses in y Direction ................................. 73 Figure 5-15 Stress Distributions of Slab at 24th Hour (Parking Structure) (MPa). (A)
Stresses in x Direction (B). Stresses in y Direction ................................. 74 Figure 5-16 Temperature Distribution of Slab of Office Building (A) 5th hour (B) 12th
hour (C) 14th hour (D) 20th hour (E) 24th hour ........................................ 76 Figure 5-17 Temperature Gradient through Thickness at Point1.......................... 77 Figure 5-18 Mean Temperature Distribution of layers of Slab (Office Structure) ... 77
Figure 5-19 Displacements at 24th Hour. (A) x Direction (B) y Direction ............... 78
Figure 5-20 Stress Distributions of Slab at 5th Hour (Office Structure) (MPa). (A)
Stresses in x Direction. (B) Stresses in y Direction ................................. 79 Figure 5-21 Stress Distributions of Slab at 12th Hour (Office Structure) (MPa). (A)
Stresses in x Direction. (B) Stresses in y Direction ................................. 80 Figure 5-22 Stress Distributions of Slab at 14th Hour (Office Structure) (MPa). (A)
Stresses in x Direction (B). Stresses in y Direction ................................. 81 Figure 5-23 Stress Distributions of Slab at 20th Hour (Office Structure) (MPa). (A)
Stresses in x Direction (B). Stresses in y Direction ................................. 82 Figure 5-24 Stress Distributions of Slab at 24th Hour (Office Structure) (MPa). (A)
Stresses in x Direction (B). Stresses in y Direction ................................. 83 Figure 5-25 Stress Distribution in x Direction at 14th Hour. (A) Parking Structure
(B) Office Structure ............................................................................... 85 Figure 5-26 Plan View of L-Shaped Structure – Wall 7 and Column 35 ................ 86 Figure 5-27 Shear Reaction Forces - Wall 7 ......................................................... 87 Figure 5-28 Shear Reaction Forces - Column 35 ................................................. 87 Figure 5-29 Parallel Performance of Solution Algorithm ....................................... 88 Figure 5-30 Parallel Performances of Initialization and Duration of One Step ....... 88
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LIST OF TABLES
TABLES
Table 3-1 Properties of 2D Finite Elements ......................................................... 29 Table 3-2 Properties of 3D Finite Elements ......................................................... 30 Table 3-3 Shape Functions for Finite Elements ................................................... 31
Table 4-1 Straight Beam with Static Loads – Model Properties .............................. 38
Table 4-2 Straight Beam with Static Loads – Load Cases ..................................... 39 Table 4-3 Straight Beam with Quad Elements (Quad4 & Quad8) – Displacements at
the Free-End ......................................................................................... 39 Table 4-4 Straight Beam with Quad Elements (Quad4 & Quad8) – Stresses at the
Fixed-End ............................................................................................. 40 Table 4-5 Straight Beam with Triangular Elements (TriM3 & TriM6) –
Displacements at the Free-End.............................................................. 41 Table 4-6 Straight Beam with Triangular Elements (TriM3 & TriM6) – Stresses at
the Fixed-End ....................................................................................... 41 Table 4-7 Straight Beam with Static Loads – Model Properties ............................. 42 Table 4-8 Straight Beam with Static Loads – Load Cases ..................................... 43 Table 4-9 Straight Beam with Hexahedral Elements (Brick8 & Brick20) –
Displacements at the Free-End.............................................................. 43 Table 4-10 Straight Beam with Hexahedral Elements – Stresses at the Fixed-End 45 Table 4-11 Straight Beam with Linear Wedge Elements (Wedge6) – Displacements at
the Free-End ......................................................................................... 46 Table 4-12 Straight Beam with Quadratic Wedge Elements (Wedge15) –
Displacements at the Free-End.............................................................. 46
Table 4-13 Straight Beam with Wedge Elements – Stresses at the Fixed-End ........ 47 Table 4-14 Straight Beam with Linear Tetrahedral Elements (Tet4) – Displacements
at the Free-End ..................................................................................... 48 Table 4-15 Straight Beam with Quadratic Tetrahedral Elements (Tet10) –
Displacements at the Free-End.............................................................. 49 Table 4-16 Straight Beam with Tetrahedral Elements – Stresses at the Fixed-End 49 Table 4-17 2D Temperature Load – Model Properties ........................................... 51 Table 4-18 2D Temperature Load – Displacements and Axial Stresses ................. 51
Table 4-19 3D Temperature Load – Model Properties ............................................ 52
Table 4-20 3D Temperature Load – Axial Stresses (Bricks and Wedges) ............... 53 Table 4-21 3D Temperature Load – Axial Stresses (Tetrahedrons) ........................ 53
Table 4-22 Square Plate Problem – Model Properties ............................................ 54 Table 4-23 Performances of Each Element ........................................................... 55
Table 4-24 Rectangular Plate Problem – Model Properties ..................................... 57
Table 4-25 Performances of Elements in Element Library ..................................... 58
Table 4-26 Performance of Heat Transfer Analysis Algorithms .............................. 59
Table 5-1 Material Properties of Parking Structure .............................................. 64 Table A-1 Integration Point Scheme for Line Element .......................................... 97 Table A-2 Integration Point Scheme for Quadrilateral Element ............................ 98 Table A-3 Integration Point Scheme for Triangular Element ................................. 98 Table A-4 Integration Point Scheme for Hexahedral Element ............................... 99 Table A-5 Integration Point Scheme for Wedge Element ....................................... 99 Table A-6 Integration Point Scheme for Tetrahedral Element ............................. 100
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CHAPTER 1
INTRODUCTION
1.1. Problem Definition
Concrete is a composite material used commonly for any type of structures such as
buildings, dams, pipes, roads, etc. due to its advantageous properties. In fact, its
higher strength in compression, workability and being cheaper than the other
construction materials are the reasons why concrete is the main construction
material all over the world.
Concrete structures generally show volume change under four different time dependent effects; elastic deformation, creep, shrinkage and temperature change.
Creep and shrinkage effects occur due to time dependent changes in the material
properties. Indeed, creep mechanism is related to change of elastic properties of
concrete with respect to time. In addition, temperature change, moisture content,
humidity and stresses on the structure affect the mechanism of the creep. Similarly, shrinkage mechanism depends on moisture content and time. Unlike
creep, it is independent of stress on the structure. Axial deformations due to
temperature changes may cause significant stresses in structures with high degree
of indeterminacy.
PTI, Post Tension Institute presents that according to ACI Committee 318, during the design of reinforced concrete structures, the aforementioned time dependent
effects should be considered in serviceability and strength conditions. It also
requires that these effects should be taken into account as reliable with the
practical applications; accordingly, instead of upper bound values, more realistic
conditions should be utilized. Current design approaches, however, use very simple
assumptions and simple analysis methods for the consideration of time dependent axial deformations of concrete structures which may lead to excessive use of
construction materials. Among the aforementioned time dependent effects,
determination of maximum and minimum temperature gradient that can occur in
the concrete structural components is always the problematic one. The common
engineering approach is to use the yearly weather temperature information and come up with design temperature difference values. Unfortunately, this approach
ignores the environmental conditions and heat transfer properties of concrete
which might produce unrealistic design forces and stresses.
Change in temperature causes volume change (elongation/shortening) of the
unrestrained structures without any stress if nonlinear thermal gradient is ignored.
On restrained structures, however, stresses are generated due to the temperature gradient. Indeed, inability of thermal expansion/shortening causes stress on the
body. Effects of temperature gradient depend on the volume of a structural
component. For structural components with relatively small volumes such as
columns, temperature change effects can be neglected. On the other hand,
temperature gradient induces significant amount of stress on components with large volumes such as slab systems. For such components’ thermal stresses can
cause significant cracking if necessary precautions are not taken.
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Since concrete has low conductivity, temperature gradient occurs among the depth
of the structure. This temperature gradient induces uneven stress distribution along the thickness of the structure. In addition, having high heat capacitance
causes the change of the heat energy in the concrete structure slowly. In other
words, the energy coming from the ambient conditions such as convection or
radiation does not affect the inside of the section immediately. Therefore, warming
and cooling of concrete structures occurs slowly and thus the temperature gradient
of the concrete is not same with the temperature gradient of ambient. In order to consider these properties of concrete structures, more detailed solution procedures
are required.
In this study, coupled, heat transfer and structural analyses, solution methods will
be developed by utilizing finite element method. This way, thermal behavior of the
reinforced concrete structure will be solved by considering the environmental thermal effects and actual temperature gradient in components of the structure will
be obtained. Then, thermal strains due to thermal gradient will be computed and
the structural solution will be performed. Accordingly, more realistic stress
distribution and internal forces due to thermal gradient will be obtained.
1.2. Related Work
According to PTI, structures having large plans with short floor to floor distance
such as parking structures are subjected to four different types of shortening.
These are shortenings in post-tension slabs, creep, shrinkage and temperature
change. In order to examine the effects of these four mechanisms, PTI analytically examined the floor shortening of a parking garage in Houston, Texas. According to
the results of the example, the largest shortening occurs due to shrinkage with the
percentage of 55%. Temperature change is the second largest shortening
mechanism whose percentage is 27.7%. The percentages of elastic and creep
shortenings are 8% and 9.3%, respectively. These results show that temperature change should not be ignored for designing of expansion joints for parking
structures (PTI).
When a reinforced concrete structure is subjected to thermal loads due to
temperature gradient, stress occurs on the structure if thermal expansion of the
structure is restrained (Vecchio, 1987). Such stresses may cause cracking of the
body. There are several techniques for represent the actual behavior of concrete structures under thermal loading. Temperature strains are calculated by
multiplying thermal coefficient of concrete with temperature change values being
maximum seasonal temperature changes for that location (PTI). This way, no
temperature gradient through section thickness is taken into account.
Iqbal (2012) stated that most parking structures are concrete, open and unheated structures. Accordingly, creep, shrinkage and temperature affect such structures
by changing volume. These effects induce displacements in structures. If these
displacements are restrained, additional stresses occurs in the structure that cause
crack, leaks and premature deterioration in the structure.
According to Iqbal (2012), in general, the duration of construction of such
structures is more than one year. In order to simplify the determination of the construction temperature, Tc, mean value of the construction season is utilized. It was stated that minimum temperature, Tmin is defined by Federal Construction
Council (Technical Report No 65) as the equal or greater than the 99% of the
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temperature of winter months at that location. Accordingly, design temperature is
calculated by utilizing the Equation 1-1.
(1-1)
According to Saetta et al. (n.d.), for structures having large concrete bodies such as
bridges or dams, temperature gradient occurs due to thermal loads; nevertheless,
the difficulties of modeling the actual environmental conditions force designers to
utilize simplified methods. This causes unreliable results. Accordingly, for analyzing such structures under thermal loads, temperature gradient occurring inside the
concrete structure should be taken into account. For linear stress calculations,
they followed three assumptions. First assumption is that heat transfer and
structural analysis are performed independently (coupled analysis in weak form).
The second one is utilizing small displacements and strains. Finally, linear elastic
material properties are considered only.
By utilizing these three assumptions, Saetta et al. (n.d.) analyzed the Sa Stria dam, built in Sardegna (Italy) and a box girder bridge section subjected to climatic
conditions of northern Italy. These structures were analyzed by utilizing coupled
analysis in weak form. First, Sa Stria dam is a roller compacted concrete dam. They
performed transient heat transfer solution to the dam by utilizing heat of hydration and heat convection conditions. This way, they obtained temperature gradient
occurring in the dam body at several days. Then, they performed linear static
analysis in order to obtain stresses on the dam body due to the temperature
gradient. According to the results, tensile stress occurred near the edges where the
temperature was lower; whereas, at core of the dam, compressive stress generated.
Similarly, Saetta et al. (n.d.) analyzed a box girder bridge section by utilizing actual thermal conditions such as heat convection, heat radiation etc. They applied
coupled analysis in weak form and obtained temperature gradient and related
stress and force distribution. According to the results, the maximum stress
occurred in wings of the section. Accordingly, they indicated that if those regions
are not designed properly, cracking may occur due to temperature gradient.
Vecchio stated that for continuous structures, thermal stress can be divided into two parts, primary and secondary thermal stresses (1987). Primary thermal
stresses occur on unrestrained structures due to nonlinear thermal gradient
through thickness. Indeed, since the thermal expansion coefficients of concrete and
reinforcing bars are not same, internal restriction occurs between concrete and
bars. Accordingly, internal stresses occur although the thermal expansion of the structure is not restrained. On the other hand, secondary thermal stresses generate
on restrained structures. According to Vecchio (1987), the secondary thermal
stresses are more critical than the first one.
Vecchio presented nonlinear frame analysis procedure for solution of reinforced
concrete frames under thermal loading. The general solution procedure is the same with the most of the linear elastic frame analysis programs. On the other hand, this
procedure provides to apply more factors such as nonlinear material, nonlinear
thermal gradient, thermal creep, time history etc. Indeed, he added the effects of
elevated temperatures to the physical and material properties such as strength or
stiffness etc. He also implemented nonlinear temperature gradient through
reinforced concrete section. He utilized the standard one dimensional heat transfer principals and he calculated the temperature values at any depth through
thickness of the structure. Vecchio compared the performance of the solution
procedure with the experimental results and obtained fair accuracy (1987).
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In another paper, Vecchio et al. (1992) stated that reinforced concrete structures
are subjected to thermal loads such as design function of the structure, ambient conditions, heat of hydration or fire. These loadings cause nonlinear temperature
and strain profiles which produce increased level of stress, distortion and damage
(i.e. primary thermal stresses). In addition to the primary thermal stresses, thermal
loads induce restrained structural deformation (i.e. secondary thermal stresses). it
was stated that second thermal stresses are more significant.
Vecchio et al. (1992) also stated that ACI Committee 349 includes less computation about the analysis of concrete structures under thermal load and this solution
technique does not represent the actual behavior. In order to investigate the
behavior of concrete shell structures under thermal load, they performed two tests.
First, they tested concrete slab under both thermal and mechanical loadings. The
slab was simply supported at each corner and concentrated load was applied to the center of the slab. In addition to mechanical load, the slab was subjected to heat at
the top surface; whereas, temperature of bottom face was kept close to room
temperature. Accordingly, temperature gradient was through the thickness. This
test was repeated for different reinforcement ratios and orientations. The
displacements at the center of the slab were compared with the analytical
solutions. They computed analytical solution by utilizing the Equation 1-2.
(1-2)
In Equation 1-2, Δc, h, l, 𝛼c, ΔT are deflection at center of slab, thickness of slab, length of slab span, thermal expansion coefficient and temperature gradient,
respectively. Since the slab was simply supported, no external stress due to
thermal loading was expected. On the other hand, since reinforcement and concrete have different thermal coefficients, reinforcing bars restrained the slab; accordingly,
internal stresses occurred in the slab body. In other words, nonlinear thermal
gradient occurred. They, however, indicated that the effects of primary thermal
stresses are negligible and no crack was occurred at the specimen during these
tests.
Second, Vecchio et al. (1992) tested the same specimen under thermal load by restraining the center along thickness direction. This restriction caused stresses
and related cracks on the slab. According to test results, internal forces increased
up to occurring of first crack. This crack causes reduce in stiffness; accordingly,
immediate relaxation occurred.
Chou and Cheng (n.d.) presented the study of measuring joint movements and seasonal thermal stresses of concrete slab located at the Chiang Kai-Shek
international airport. They used optical fiber sensors to measure the joint
displacements due to seasonal temperature change. These sensors were located at
the middle layer of the slab through thickness and for approximately one year,
displacements and temperature values had been stored. Since the sensors received
temperature of only middle point layer, stresses on the slab were calculated with the assumption of constant temperature change along the thickness. They
calculated stresses by considering the shrinkage mechanism also. According to
their results, tensile stress will occur on concrete slab most of the time due to
temperature changes if the casting of it is performed in hot temperatures. In
addition to this, they made predictions about the future movement and thermal
stresses by utilizing regression analysis.
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5
According to Li et al. (2009), concrete slab bends if it is subjected to negative
temperature gradient through thickness. This bending causes tensile stress at top layer of the slab. Although the maximum tensile stress is expected at the bottom
layer of the slab, due to negative temperature gradient, it can occur at top layer.
Accordingly, first cracking occurs at top layer. Because of this, they solved the
system by acting the thermal and axle load together.
Li et al. (2009) utilized linear temperature gradient through thickness of the slab
and solved the slab by considering each combination of axle load. This solution was performed by utilizing finite element method (FEM). According to their study, the
maximum tensile stress occurred at top layer of the slab. This causes cracking from
top to bottom; although, the designers in China expected cracks from bottom to
top. On the other hand, these results are obtained from a structural model
composed of a single slab. Behavior of an indeterminate system was not considered.
Thelandersson stated that thermal loading can be added to the mechanical analysis as initial strain for both linear and nonlinear solutions (1987). According to
Thelandersson, this approach was developed for metals; whereas, for concrete, the
mechanism is more complex than stated above because mechanical properties
depend on temperature. Accordingly, Thelandersson stated change of strain by
utilizing the Equation 1-3.
̇
̇
̇ (𝛼 ) ̇ (1-3)
Where
(
) (1-4)
(
) (1-5)
1 and 2 represent change of elastic properties of concrete with respect to temperature and if these elastic properties are independent of temperature,
isotropic linear thermo-elastic material behavior is obtained (Thelandersson, 1987).
Thelandersson developed constitutive material model including the derivatives of elastic properties and verified the method with experimental results. According to
the results, this tangent modulus gives reliable results; although, it is simple.
Borst and Peters presented the material behavior of concrete under elevated
temperatures (1988). Indeed, they indicated that concrete behaves nonlinear under
elevated temperature due to thermal dilatation, temperature dependent material
properties, transient creep, and cracking. Transient creep mechanism includes both thermal expansion with thermal expansion coefficient whose function is
nonlinear of the temperature and change of elastic properties of the material such
as Modulus of elasticity etc.
According to Borst and Peters (1988), large scale structures should be solved by
utilizing smeared crack formulation. Otherwise, reliable results are not obtained. However, derivation of the material behavior composed of smeared crack and other
nonlinear mechanisms stated above is not appropriate. Accordingly, they utilized
strain decomposition approach to handle this problem. Indeed, they used separate
constitutive law for each strain rate. They simulated the test of plain concrete
cylinders. They stated that this test was conducted by Anderberg and
Thelandersson (1978) in order to discover the mechanical behavior under high
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6
temperatures. This simulation reveals that analysis of concrete structure does not
represent the actual behavior if transient creep is not taken into account.
1.3. Objectives and Scopes
The main objective of this study is the development of finite element solution
platform that enables the coupled, thermal and structural, analysis of civil
engineering structures. This allows the detailed investigation of stress caused by
thermal effects in any structure of any geometry.
For this purpose, linear heat transfer and linear structural analysis solution
algorithms are combined as weak form of coupled analysis. Since for structural solutions, no significant geometry change is generated, thermoelastic property of
the material may be ignored. Accordingly, weak form of coupled analysis is
preferred. In other words, linear heat transfer analysis and linear structural
analysis are performed in sequential order. Heat transfer analysis computes
temperature values of certain locations at a body for a certain time period and structural analysis uses these temperature values for calculating the thermal
strains. These thermal strains are then converted to equivalent nodal forces and
the corresponding deformations are computed.
In addition, several types of 2D and 3D finite elements will be developed. This way,
structures with complex geometries can also be analyzed. Each element will include
both structural analysis and heat transfer solution related algorithms. Moreover the performance of each element for both cases for several benchmark problems will be
investigated.
By utilizing the developed solution algorithms, the top floor of a typical L-shaped
building will be analyzed as a case study. Indeed, the building will be solved twice
with different thermal conditions. First, it will be analyzed as a parking structure being open and subjected to ambient temperature conditions only. For the second
case, the same building will be analyzed as an office building thus the internal
temperature of the building will be fixed to 20OC. Both structures will be subjected
to heat convection with ambient temperature of Adana at July 23rd (Bulut et al.,
n.d.) but the heat radiation effect will be ignored. Also, heat convection occurring
on the columns and walls are neglected. Casting temperature of the structure is assumed to be equal to 14OC and temperature gradient of slab through thickness
for only one day will be investigated. For several hours of that day, stresses on the
slab due to temperature gradient at that time will be calculated and compared with
each other.
For both the heat transfer and structural analyses, all material properties will be assumed as linear. In other words, effects of nonlinear stress - strain relationship,
nonlinear temperature gradient, transient creep, and shrinkage will be ignored.
Only stresses generated due to temperature change will be discussed.
1.4. Thesis Outline
Outline of the thesis is as follows. Theory of solution methods of solid mechanics
and heat transfer and finite element procedures are discussed in Chapter 2. All
implementations are presented in Chapter 3. In this chapter, structures of solution
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7
algorithms, linear structural and linear heat transfer analysis and coupled system
are presented. In addition, finite elements existing in finite element library of the platform and their basic properties are explained. Chapter 4 includes verification of
the solution algorithms and finite elements stated in Chapter 3. Behaviors of L-
shaped concrete structures under different thermal loads are going to be discussed
in Chapter 5. Indeed, parking and office structures having same geometry and
different thermal loading conditions are compared. Finally, Chapter 6 is conclusion
part of the thesis. In addition to these, properties of integration points used for
calculation of integrals numerically are tabulated in the appendices.
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8
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9
CHAPTER 2
THEORY
2.1. Introduction
In this chapter, theories used for this study are explained. Theory of solution
algorithms, structural, heat transfer and coupled analysis equations and finite
element method are discussed. Indeed, general equations of each solution method, structural analysis and heat transfer, and adapting them to finite element method
are explained briefly. Moreover, theory of the coupled analysis procedure derived by
utilizing these heat transfer and structural analysis solution is discussed.
2.2. Structural Analysis
Structural analysis solution is derived from principals of thermodynamics with the
assumption of having uniform and constant temperature distribution over the
body. The strong form of general mechanical equation second order differential
equation is presented in Equation 2-1.
̈ (2-1)
In Equation 2-1, ρ, C, b, and u indicate density, constitutive material matrix, body
load and displacement, respectively. This equation can be rewritten in Galerkin functional form by using integration by parts and Gauss integral theorems
(Equation 2-2).
( ) ∫ ̈
∫ ( )
∫
∫
(2-2)
In Equation 2-2, u represent test function and it is zero at boundary. According to finite element discretization yields the following expressions.
̈ ̈
̈ ̈ (2-3)
In Equations 2-3, d represents the element displacement vector and N is shape
function. Inserting definitions (Equation 2-3) into the Galerkin functional yields to
the Equation 2-4.
∫ ̈
∫
∫
∫
(2-4)
By rearranging the Equation 2-4, Equation 2-5 is obtained.
∑ (∫
̈
∫
∫
∫
) (2-5)
Equation 2-5 may be stated in terms of internal and external forces (Equation 2-6).
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10
∑ ( )
∫ ̈
∫
∫
∫
(2-6)
For arbitrary test function the following equation (Equation 2-7) should be satisfied.
(2-7)
The equation 2-6 implies that internal forces are function of nodal displacements,
d.
̂( ) (2-8)
Therefore, internal forces may be expressed in matrix form (Equation 2-9).
̈
∫
∫
(2-9)
and stated in Equation 2-9 are element mass and stiffness matrices, respectively. In Equation 2-9, strain-displacement relation matrix is indicated with
letter B and calculated from Equation 2-10.
[
]
(2-10)
Equation 2-10 is valid for 3D finite elements; whereas, B matrix of a 2D element has two rows. In other words, it includes derivative of shape functions with respect to two axes. In this equation, number of shape functions indicated with letter of m.
Matrix form of general structural analysis equation (Equation 2-2) is presented in
Equation 2-11.
̈ (2-11)
In Equation 2-11, M, K, F and d are mass and stiffness matrices and external force
and displacement vectors, respectively. For linear static solution, time derivative of
the displacement is zero. Accordingly, Equation 2-11 is simplified and general
linear static equation (Equation 2-12) is obtained.
(2-12)
In the Equation 2-12, since the system is linear, stiffness matrix is computed by
using initial geometry and linear material properties. The external force vector can
be calculated as Equation 2-13.
(2-13)
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11
External force vector can be divided into two parts, nodal loads and element loads
(Equation 2-13). Nodal loads come from input directly. On the other hand, element loads are calculated by using the element geometry and material properties and
converted to equivalent nodal loads. There are four different types of element loads;
body load, surface load, thermal load, initial strain load presented in Equations 2-
14 to 2-17, respectively.
∫
(2-14)
∫
(2-15)
∫ ( )
(2-16)
∫
(2-17)
In Equations 2-14 to 2-17, W, Δt, a and ε represent uniform surface load,
temperature change, thermal coefficient and strain vector, respectively.
In Equation 2-16, thermal load due to constant temperature change over body is calculated. Whereas, since in coupled analysis, nodal temperature change values
are obtained from heat transfer solution, modification of calculation of thermal load
is required. In fact, thermal strains and corresponding stresses due to temperature
change at each nodal point are calculated (Equations 2-18 and 2-19). Then, these nodal stress values contribute to calculation of equivalent nodal load with the rate
of weight value of the corresponding integration points. Numerical integration of
calculation of equivalent load vector is presented in Equation 2-20.
(2-18)
(2-19)
∑ | |
(2-20)
In equations 2-18 to 2-20, σ and w are stress vector and weight value of the
integration point scheme.
Each integral is handled numerically by utilizing Gauss Quadrature rule. Some
integration point schemes cause problematic element behaviors such as shear/
membrane locking or hourglassing modes etc. First, shear/membrane locking
occurs in linear elements if full integration scheme is utilized (Dhondt, 2004). As a matter of fact, under pure bending load case, there is no shear strain in the body
since no shear force exists. However, if full integration scheme is used, virtual
shear strains occur at gauss points existence of shear strain makes the behavior
stiffer. On the other hand, utilizing reduced integration scheme hinders generation
of shear locking since virtual shear strain does not occur at center of the element.
The other problematic behavior is hourglassing mode called also zero energy mode. It occurs if displacement modes of element do not create any strain and stress at
the integration points (Dhondt, 2004). Presence of this problematic behavior can be
checked by using the Equation 2-21 (Dhondt, 2004).
( ) ( ) (2-21)
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12
In Equation 2-21, nZEM, nd, n, nIP, nS, and nRBM represent numbers of zero
energy modes, degree of freedoms, nodes, integration points, strain components
and rigid body modes, respectively. As seen in the equation, this problem can be
handled by increasing the number of integration points.
2.3. Heat Transfer Analysis
The heat transfer occurs due to energy transfer between material bodies because of
their temperature difference (Lewis et al., 2004). There are three different ways of
energy transport:
Conduction
Convection
Radiation
Conduction
Conduction mode occurs by transporting energy from one molecule to another without any motion of these molecules (Lewis et al., 2004). Therefore, conduction
mode heat transfer occurs between solid bodies.
This mode can be explained by Fourier’s law. The transferred energy per unit time
and per unit area is presented in Equation 2-22.
( ) (2-22)
In Equation 2-22, q, k, and divergence of represent heat flux (W/m2), thermal conductivity (W/mOK) and temperature gradient (OK/m), respectively.
Convection
Convection mode comes into existence by transferring energy from one molecule to
another with free motion of molecules belonging to liquids or gases (Lewis et al.,
2004). Because of this, heat transfer between a solid and fluid can be described by
heat convection. There are two types of convection; forced convection and free convection. In forced convection, fluid is sent to the solid material with an external
force such as pump or fan; whereas, there is no external contribution in free
convection.
Convection heat transfer can be described by Newton’s law of cooling. The
transferred energy per unit time can be calculated with Equation 2-23.
( ) (2-23)
In Equation 2-23, convection heat transfer coefficient (W/m2 OK) and temperature
difference between body and fluid (OK) are symbolized with h and θ-θa, respectively.
The direction of heat flux stated in Equation 2-23 is perpendicular to the boundary.
Radiation
Lewis et al. states that the radiation occurs in all bodies at all temperature. In fact,
all bodies transfer their energy by emitting radiation (2004). Because of this, it is
not required to contact between bodies to change their temperatures. When the
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13
radiation waves emitted by a body hit to surface of another body, some of these
waves are reflected, some part is transmitted, and the remaining part is absorbed
(Lewis et al., 2004).
Stefan – Boltzmann law is related with radiative heat transfer mode. The
transferred energy per unit time is found by Equation 2-24.
(2-24)
In Equation 2-24, θ and σ are surface temperature (OK) and Stefan – Boltzmann
constant (5.669*10-8 W/m2 OK4), respectively. Similar to heat convection, the
direction of heat flux is perpendicular to the boundary.
Formulation of Heat Transfer
Total energy in current direction is calculated by multiplying the flux with the
perpendicular area (Equation 2-25).
(2-25)
According to conservation of energy law, energy storage in a system is equal to the
difference between the inlet energy and outlet energy. Conservation of energy law is
displayed in Equation 2-26.
(2-26)
Figure 2-1 Control Volume (Lewis et al., 2004)
In Figure 2-1, control volume of a body and inlet/outlet heat energies are represented. The output energies can be redefined by substituting Taylor Series
expansion without higher terms (Equation 2-27).
(2-27)
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14
Moreover, the heat generation and rate of energy storage of control volume are
presented in Equations 2-28 and 2-29, respectively.
(2-28)
(2-29)
In Equation 2-28, G is rate of internal heat generation per unit volume (W/m3).
Similarly, in Equation 2-29, ρ and cp indicate density and specific heat,
respectively. Substituting Equations 2-27, 2-28 and 2-29 into conservation of
energy equation (Equation 2-26) yields the general equation of heat transfer
(Equation 2-30).
(2-30)
Simplified form of Equation 2-30 is presented in Equation 2-31.
̇ (2-31)
In Equation 2-31, qc, qC, and qr are conduction, convection and heat radiation
fluxes, respectively. Similar to structural analysis equation, Equation 2-31 can be solved by utilizing Galerkin functional. Galerkin functional form of the general heat
transfer equation is presented in Equation 2-32.
( ) ∫
∫
∫ ∫ ̇
(2-32)
In Equation 2-32, is test function which is zero at boundaries and surface flux, qs includes both heat convection and heat radiation fluxes. By utilizing integration
by parts and Gauss integral theory, rearranged form of Equation 2-32 is obtained
(Equation 2-33).
∫ ( )
∫
∫
∫
∫ ̇
(2-33)
Finite element discretization of heat transfer equation yields to the following
expressions.
̇ ̇
̇ ̇ (2-34)
In Equations 2-34, T represents the element temperature vector and N is shape
function of the element. Inserting definitions stated above into the Galerkin
functional yields to the Equation 2-35.
∫
∫ ( )
∫ ( )
∫
∫ ̇
(2-35)
Equation 2-35 can be written in matrix form (Equation 2-36).
̇ (2-36)
In Equation 2-36, Ce and Ke are element heat capacitance and thermal stiffness
matrices and they are presented in Equations 2-37 and 2-38, respectively.
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15
∫
(2-37)
∫
∫
(2-38)
In Equation 2-38, stiffness matrix includes conduction and convection modes first and second term, respectively. Thermal forces, however, do not depend on element
temperature vector. It is possible to separate element thermal forces into groups
such as heat generation, heat convection, heat radiation and surface flux forces
(Equation 2-39).
∫
∫
∫ ( )
∫
(2-39)
In this study, radiation part was ignored; accordingly, there are three different
thermal loadings, heat generation, heat convection, and surface flux (Equation 2-
39). The direction of surface flux loading is inward to the body.
There are two boundary conditions of differential equation of heat transfer physic; constant nodal temperatures and surface flux. Indeed, constant temperature and
surface flux are essential and Neuman boundary conditions, respectively (Equation
2-40). Surface flux boundary condition can include heat convection, heat radiation,
and external flux.
Essential BC:
Neuman BC: ( ) (2-40)
2.4. Coupled Analysis Methods
The term coupled analysis refers to the combined analysis of multi-physics
problems. There are two ways of coupled solution, strong and weak formulation.
In strong form, different physics analyses are performed at the same time.
Therefore, each effect of these analysis types is included in solution of the problem. Strong form of coupled analysis equation in matrix form is presented in Equation 2-
41.
[
] {
} {
} (2-41)
In equation stated above, X1 and X2 represent solution vectors of two different physics. K and F are stiffness matrix and force vector, respectively. Indeed, first and
second rows indicate different types of analysis systems. These two analysis
systems are connected to each other due to K12 and K21 matrices.
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16
On the other hand, in weak form of coupled analysis, different physics analyses are
performed, sequentially. In other words, first one analysis system is performed and results of it implemented to the other solution system. Then the second analysis is
executed. The matrix form of coupled analysis in weak form is presented in
Equation 2-42.
[
] {
} {
} (2-42)
Unlike strong form, off diagonal terms are zero matrices in weak form solution. This
makes these analysis systems unbounded. However, force vector of second analysis
system includes not only external load of it but also loads due to solution of first
equation. This loading is the only connection between these equations.
Due to having unbounded stiffness matrix, weak form requires less memory during
execution and implementation of it is easier than the strong form.
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17
CHAPTER 3
IMPLEMENTATION
3.1. Introduction
In this chapter, implementation of solution algorithms and finite elements used for
this study are discussed. In fact, linear static analysis algorithm, linear heat
transfer algorithms (steady-state and transient) developed for this study, and coupled analysis system obtained by executing heat transfer and structural
analysis algorithms sequentially, are explained. In fact, procedure of linear static
and linear steady state algorithms solve linear system of equations. On the other
hand, time integration schemes are used for solution of transient heat transfer
algorithm. Finally, for coupled analysis, either steady-state or transient heat
transfer solution and linear static algorithm are performed, sequentially.
Moreover, several two and three dimensional finite elements were developed. Each element has two formulation types, linear and quadratic and is suitable for two
different physics problems, structural analysis and heat transfer. In structural
analysis part, elements have routines required for linear static analysis such as
calculation of linear stiffness matrix, internal force and element stress vectors. Moreover, they have the capability of converting element loads such as body load,
surface load, temperature differences and initial strain to equivalent nodal loads. In
addition to structural analysis part, each element has linear heat transfer analysis
routines such as calculating linear conduction, heat capacitance and various types
of loading such as heat convection, surface flux, and heat generation.
3.2. Structure of Panthalassa
For implementation of the new solution algorithms and finite element models, a
finite element analysis platform, Panthalassa was used (Kurç et al., 2012).
Panthalassa is an extensible finite element analysis environment which was developed by using C++ language with object-oriented data structure (Bahçecioğlu
et al., 2012). Panthalassa includes an analysis engine that performs data
input/output, handling of the structural objects, such as finite elements, loading
definitions etc., and general routines such as matrix assembly, solution etc. the
design of the engine allows addition of new solution algorithms, material models or
finite elements externally as plug-in modules.
In this study, owing to extensibility property of the platform, several two and three dimensional finite element models and linear heat transfer solution algorithms were
developed and added to the platform in the plug-in format. In fact, Panthalassa has
virtual classes such as element, material model and solution algorithm etc. and
these virtual classes let user to develop a new class including same properties with them and be implemented to the platform in plug-in format (Kurç et al., 2012).
Because of this, heat transfer analysis plug-in having linear steady-state and
transient solution algorithms can reach the model properties such as loading and
boundary conditions from the platform and give the results to it. Similarly, each
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18
finite element reaches the geometry and material property of the system and gives
the element matrices such as stiffness or stress etc. This process is illustrated in
Figure 3-1.
Engine of Panthalassa
Solution Algorithm
Plug-in
Input
Output
Model Properties
Analysis Results
Finite Element Plug-in Element
Properties
ModelProperties
Figure 3-1 Connections of Plug-ins with Panthalassa Engine
3.3. Solution Algorithms
In this section, structure and implementation to the Panthalassa platform of solution algorithms, linear static and linear heat transfer (steady-state and
transient) and coupled analysis with these two solutions are discussed.
3.3.1. Linear Static Analysis
In the linear static analysis for structural analysis problems, basically the equation
system presented in Equation 3-1 is formed and solved. For the solution of the
linear system of equations, LU decomposition method stated in MUMPS library is
used (Kurç et al. 2012).
(3-1)
In Equation 3-1, K, F, and U indicate stiffness matrix, nodal force vector, and nodal
displacement vector, respectively. Nodal force vector includes both external nodal load and equivalent nodal loads due to element loads. Equivalent nodal loads of
each element are computed by the subroutines of the element plug-ins and
assembled by the subroutines of the solution algorithm plug-in utilizing the service
routines of the Panthalassa Engine.
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19
Element Load Vector(Body + Surface + Temperature + Strain)
Element Stiffness Matrix
(Mechanic)
Finite Element
System Stiffness Matrix
Model
System Load Vector
Solution
Output
Assembly
External LoadVector
Figure 3-2 Flow Chart of Linear Static Analysis Algorithm
In Figure 3-2, flow chart of linear static analysis algorithm is presented. In fact, element stiffness matrix and equivalent element nodal load vector are computed by
each finite element and they are assembled into to the system stiffness matrix and
system nodal force vector. Such assembly operations are handled by Panthalassa
routines automatically; whereas, the element loads computations are performed at the algorithms of the plug-ins. As the stiffness matrix and the force vector of the
whole structure are obtained, they are solved by the LU decomposition based solver
routines of Panthalassa and the nodal displacements are computed. By using the
element nodal displacements, element stresses and forces are calculated. As a final
step, nodal displacements and element stresses are written to the output file for
post processing.
3.3.2. Linear Heat Transfer Analysis
The basic equation for the general heat transfer problem in matrix form is
presented in Equation 3-2.
̇ ( ) (3-2)
In the above equation, heat capacitance matrix, thermal stiffness matrix, thermal load vector, and nodal temperature vector are represented by letters C, Kt, Ft(t), and
, respectively.
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20
The general heat transfer equation can be solved in two ways. In the first approach,
called steady-state solution, time derivatives of temperatures are ignored. This way the solution of the equation is significantly simplified. This solution way gives the
final equilibrium condition of the structure under given loads. Therefore, it is not
possible to obtain time required for equilibrium condition like linear static analysis.
Since the solution performs only once, duration of solution is not significant. On
the other hand, linear transient solution uses time integration scheme for solution.
Since the structure is solved for each time step, it consumes more time than the steady state way. However, transient solution calculates the behavior of the
structure even if equilibrium condition has not been satisfied yet. Since this
solution needs heat capacitance matrix, C and essential boundary conditions, more
memory is required.
Linear Steady-State Analysis
Steady-state analysis system assumes no change of temperature with respect to
time. Therefore, Equation 3-2 is simplified and general steady-state heat transfer analysis equation is obtained (Equation 3-3). In the linear steady-state analysis
approach, Kt does not change with respect to temperature values.
(3-3)
In the computational point of view, steady-state heat transfer analysis equation is similar to the linear static analysis equation (Equation 3-1). On the other hand,
forming the thermal stiffness matrix and thermal load vector is quite different than
the linear static analysis. First of all, heat stiffness matrix is composed of
conduction and convection stiffness matrices. Conduction stiffness matrix is
calculated by using material and geometric properties of the element; whereas, heat convection loading on the element influences the convection stiffness matrix in
addition to the material and geometric properties of the element. Heat convection
surface load also contributes to nodal load vector with ambient temperature. Flow
chart of matrix assembly of heat transfer analysis algorithm is shown in Figure 3-3.
Conduction Stiffness Matrix
Convection Stiffness Matrix
Heat Stiffness Matrix
Geometry of Element
Surface Load(Heat Convection)
Material of Element
Equivalent Nodal Load Vector
Ambient Temperature
Figure 3-3 Flow Chart of Matrix Assembly of Heat Transfer Solution Algorithm
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21
Three different boundary conditions, constant temperature, heat convection, and
surface flux can be described for heat transfer problems. Constant temperature condition is defined at nodal points and taken into account as restraints. Whereas,
heat convection and surface flux conditions need boundary definition. For
Panthalassa, boundary of an element is defined by giving the element nodal ids of
that boundary.
Element loads of heat transfer problems, heat convection, surface flux, and heat
generation, are handled by element plug-ins and sent to the algorithm plug-in as element equivalent nodal load vector. Unlike linear static solution, system load
vector is composed of only element loads since there is no nodal load definition for
this type of problems.
In other words, the general process of the algorithm is similar with the one of linear
static analysis except the assembly of the system matrices. Flow chart of linear
steady-state heat transfer solution algorithm is displayed in Figure 3-4.
Element Thermal Load Vector(Convection + Surface Flux + Heat Generation)
Element Thermal Stiffness Matrix
(Conduction + Convection)
Finite Element
System Thermal Stiffness Matrix
Model
System Thermal Load Vector
Solution
Output
Assembly
Figure 3-4 Flow Chart of Linear Steady-State Analysis Algorithm
As seen in Figure 3-4, general solution of the steady-state heat transfer algorithm is very similar to the one in linear static analysis algorithm. In fact, thermal stiffness
matrix and nodal load vector of elements are obtained from finite element plug-ins
and they are assembled into the system stiffness matrix and system load vector,
respectively. These system matrix and system vector are then solved by using LU decomposition method and as a result, nodal temperatures and element fluxes are
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22
obtained. Nevertheless, the only difference from linear static analysis algorithm is
the formation of system thermal stiffness matrix and thermal load vector. In linear static analysis algorithm, loadings whether element or nodal has contribution to
the nodal load vector, only. Whereas nodal load vector includes only element
loadings.
Linear Transient Analysis
The linear transient analysis is performed by utilizing two different time integration
approaches; implicit and explicit Euler. As a first step, Taylor Series expansion of
the general heat transfer equation (Equation 3-2) was calculated as shown in
Equation 3-4.
( ) ( ) ( ) ( ) ( )
( ) ∑
( )( )
( )
(3-4)
In Equation 3-4, ignoring the higher order terms yields to Equation 3-5.
( ) ( ) ( ) ( ) (3-5)
By adapting Equation 3-5 to the general equation of heat transfer, Equation 3-6 is
obtained.
( ) ̇ ( ̇) (3-6)
In Equation 3-6, , n, and indicate nodal temperature vector, number of step, and time increment, respectively. Moreover, β is the coefficient used for selecting
the solution method. In fact, the main point is to decide which slope, ̇ or ̇ is used. In here, different integration schemes having different slope definition such
as forward, backward or central difference can be taken into account by changing
β. In fact, for backward and forward Euler schemes, β is taken as 1 and 0,
respectively. Backward integration scheme use the time derivation of temperatures at current time step. Since, the slope and temperatures at current time step (n) are
not known, the method is called implicit. Whereas, forward integration scheme is
called explicit since the slope in previous time step (n-1) is required. Therefore, only
the temperature values of current time step is unknown. In addition to this, it is
possible to use any other integration scheme by inserting appropriate coefficient, β.
Substituting Equation 3-6 into Equation 3-2 gives the following equation.
( ) ̅ ( ) (3-7)
In Equation 3-7, ̅ is total nodal force vector and calculated as stated in Equation 3-8.
̅ ( ) (3-8)
In linear transient heat transfer analysis algorithm, Equation 3-7 is solved. Flow
chart of linear transient analysis algorithm is presented in Figure 3-5. In addition
to thermal stiffness and thermal load vector, heat capacitance matrix, C is also calculated at the element level and assembled to the system matrices. These system
matrices and temperature vector in previous time step (n-1) are solved by using
implicit or explicit Euler integration scheme and temperature vector of the current
time step (n) is obtained. Temperature vector of previous time step is updated and
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solution process repeats until the solution time is equal to end time. Nodal
temperature vectors in any step are saved to the output file of the model.
Element Heat Load Vector(Heat Convection + Surface Flux + Heat
Generation)
Element Heat Stiffness Matrix
(Conduction + Convection)
Finite Element
System Heat Stiffness Matrix
Model
System Heat Load Vector
Solution
Temperatures(n+1)
Output
Previous Temperatures
IFn = 1
IFn > 1
IF time < End time
Updating Variables YES
n: Time step
Assembly
Element Change of Energy Storage
Matrix
System Change of Energy Storage
Matrix
FinalizeNo
Figure 3-5 Flow Chart of Linear Transient Heat Transfer Analysis Algorithm
As stated above, implicit and explicit time integration schemes use the same equation (Equation 3-7) with different β coefficients. Taking β as zero (explicit
scheme) and lumped heat capacitance matrix, C reduces the computational cost of
inverse process. Whereas, even if lumped heat capacitance matrix, C is used,
summation with thermal stiffness matrix damages the lumped property. This
causes higher computational cost.
3.3.3. Coupled Analysis
The term coupled analysis refers to the combined analysis of multi-physics problems. The combination of different physics equations can be done by utilizing
either strong or weak forms of the governing differential equations. Matrix form of
weak form of coupled analysis is shown in Equation 3-9 (ANSYS, 2009).
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[
] {
} {
} (3-9)
In Equation 3-9, first and second row indicate heat transfer and structural analysis
solution equations, respectively. Structural force vector, includes both mechanical force and thermal force coming from heat transfer solution. Thus, it is
required to solve heat transfer and structural analysis equations, sequentially.
In the weak form, first the analysis of a single physics problem, in this case heat
transfer analysis, is performed and then the analysis of the second physics problem
(linear static analysis) is conducted utilizing the output of the first analysis.
In this implementation, first of all, transient heat transfer analysis is performed and nodal temperature values for each time step are calculated. Then, subtracting
output temperature values from initial ones, temperature change values are
obtained for static analysis. Then, these values are inserted the linear static
analysis algorithm; accordingly, nodal displacements and element stresses are obtained and saved to output file of the model. This procedure repeats until the
solution time is equal to end time. Flow chart of coupled analysis implementation
with transient solution is presented in Figure 3-6.
Nodal Temperatures
Model
Nodal Displacements and Element Stresses
Output
Linear Heat Transfer AnalysisPlug-in
Linear Static Analysis
If Time < End Time
Update Parameters Yes
Temperature Change Vlues
Finalize
No
IfTime = TimeStr
Yes
No
Figure 3-6 Flow Chart of Coupled Analysis Algorithms with Transient Solution in
Weak Form
As heat transfer analysis, it is possible to use linear steady-state solution algorithm, also. Since this algorithm does not include iterative solution,
implementation is quite simpler than the transient one. In fact, nodal temperatures
and temperature change values are calculated once and then first solution process
is finalized. The second process is same with the transient solution. This
implementation process of steady-state solution is presented in Figure 3-7.
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Nodal Temperatures
Model
Nodal Displacements and Element Stresses
Output
Linear Heat Transfer AnalysisPlug-in
Linear Static Analysis
Temperature Change Values
Figure 3-7 Flow Chart of Coupled Analysis Algorithm with Steady-State Solution in
Weak Form
Linear static analysis algorithm sends nodal temperature change values to the finite element plug-in and equivalent nodal loads go back to the analysis plug-in. In
fact, in finite element plug-ins, thermal strains due to nodal temperature change
are calculated and these strains are converted to equivalent nodal force. These
forces are sent back to the solution algorithm (linear static analysis algorithm) and
assembled into the system nodal load vector. Flow chart of the process stated above
is presented in Figure 3-8.
Nodal Temperatures
Linear Static Analysis
AlgorithmPlug-in
Finite Element Plug-in
compute Element
Load Function
Calculating Equivalent Nodal
Force due to Nodal Temperatures
Solution
1 2
Figure 3-8 Flow Chart of Converting Nodal Temperatures to Equivalent Nodal Force
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3.3.4. Parallel Solution Algorithms
Panthalassa has ability to execute parallel solution algorithms. Accordingly, parallel linear static and linear heat transfer analysis algorithms (steady-state and
transient) were developed.
Linear static and linear steady-state analyses were parallelized by utilizing MUMPS
(Multifrontal Massively Parallel Sparse Direct Solver) library (Amestoy et al., 2000).
Indeed, in this study, solution of sparse matrix was performed by utilizing MUMPS.
In these algorithms, sparse stiffness matrix is obtained at each core. MUMPS
divides the sparse matrix and distributes each sub-matrix to all cores. Then the
linear system is solved by MUMPS and the solution vector is sent to the main core. The flow chart of linear steady-state heat transfer analysis algorithm is presented in
Figure 3-9. The procedure is the same for linear static analysis algorithm.
Element Thermal Load Vector(Heat Convection + Surface Flux + Heat
Generation)
Element Thermal Stiffness Matrix
(Conduction + Convection)
Finite Element
Model
System Thermal Load Vector
Temperatures
Output
Assembly
Linear Solution of Equations
SystemThermal Stiffness Matrix
Figure 3-9 Flow Chart of Parallel Steady-State Heat Transfer Algorithm
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For parallelization of linear transient heat transfer analysis algorithm, explicit Euler
scheme was utilized assembly of the system equations and solution is performed at
the element level. Accordingly, it is very suitable for parallelization. Indeed, in explicit scheme, heat capacitance matrix was taken as lumped; accordingly, taking
inverse of lumped matrix does not cause significant computational cost. In this
algorithm, thermal stiffness matrix is divided into sub-matrices and distributed to
each core. Since each core knows the load vector and heat capacitance matrix, they
solve the each substructure. Then they transfer the solution to each other;
accordingly, each core has the total solution vector of the system. Each core updates the temperature values and repeats this procedure up to end time is
reached. The flow chart of parallel transient heat transfer analysis algorithm is
presented in Figure 3-10.
Figure 3-10 Flow Chart of Parallel Transient Heat Transfer Analysis Algorithm
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3.4. Finite Elements
In this section, details of the two and three dimensional finite elements developed for this study are discussed. Linear and quadratic formulations of quadrilateral and
triangular membrane elements were implemented as two dimensional elements.
Similarly, linear and quadratic forms of hexahedral, wedge, and tetrahedral
elements were implemented as three dimensional elements.
3.4.1. Geometrical Properties of Finite Elements
Description and general properties of 2D and 3D element are presented in Tables 3-
1 and 3-2, respectively. In element geometry columns of the table, isoparametric and Cartesian geometry of the element are presented, respectively. Similarly,
isoparametric boundary geometry of that element is listed in boundary geometry
section. Moreover, shape functions of each finite element are presented in Table 3-
3.
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Table 3-1 Properties of 2D Finite Elements
Finite
Element
Element
Description
Number
of Nodes
Degrees of Freedoms Element Geometry Boundary Geometry
Quad4
2D
Linear
Quadrilateral
4
Mechanical: Ux, Uy
Heat Transfer:
Quad8
2D
Quadratic
Quadrilateral
8
Mechanical: Ux, Uy
Heat Transfer:
TriM3
2D
Linear Triangular
Membrane
3
Mechanical: Ux, Uy
Heat Transfer:
TriM6
2D
Quadratic
Triangular
Membrane
6
Mechanical: Ux, Uy
Heat Transfer:
29
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Table 3-2 Properties of 3D Finite Elements
Finite
Element
Element
Description